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Hi everyone. This is my first post here and I was wondering if any of you could help me.

The question is to prove that

[tex]1 + \frac{x}{3} - \frac{x^2}{9} < (1 + x)^\frac{1}{3} < 1 + \frac{x}{3} [/tex] if x>0.

The question is in a section on the lagrange remainder theorem. The fact that the first expression is less than the third is self evident but I'm assuming to show that the middle expression falls between the two involves using the lagrange remainder theorem on that expression. I don't really know how exactly I would go about squeezing it between the two expressions, however, so I'm looking for some pointers on that.

Thanks!

Edit: Nevermind. I was able to figure it out by using the first and second degree taylor polynomials of the second expression and determining sign of the remainder. Thanks though!

The question is to prove that

[tex]1 + \frac{x}{3} - \frac{x^2}{9} < (1 + x)^\frac{1}{3} < 1 + \frac{x}{3} [/tex] if x>0.

The question is in a section on the lagrange remainder theorem. The fact that the first expression is less than the third is self evident but I'm assuming to show that the middle expression falls between the two involves using the lagrange remainder theorem on that expression. I don't really know how exactly I would go about squeezing it between the two expressions, however, so I'm looking for some pointers on that.

Thanks!

Edit: Nevermind. I was able to figure it out by using the first and second degree taylor polynomials of the second expression and determining sign of the remainder. Thanks though!

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